How do i fit those eyes into my character mesh?
Both your eye globe and your eye hole have irregular shape, so if you want to solve your problem you need to make them both as regular as possible.
Another way to create your eye would be not to move the eye itself, but to move the texture on the surface of the eye, using an empty as Texture Coordinate Object, like this:

Here is the material setup:

You could get a similar result with the UV Warp modifier as suggested by Lemon.